A reflex is an unconcious response to a potentially harmful stimulus, such as pulling a tray out of the oven without oven gloves on!
Reflexes bypass the brain and instead have a circuit that only passes through the spinal cord. This means reflexes are very quick (as they don't travel far), which is useful as they function to protect us.
A reflex arc is the name we give to the pathway of neurones from detecting the stimulus to initiating a response. It is conveniently shaped like an arc, travelling to the spinal cord and back again.
Going back to our example about touching a hot tray, this is first detected by thermoreceptors in the skin that detect heat changes (thermo = heat). These send a signal down a sensory neurone that travels to the spinal cord. Here, it synapses (connects to) with a relay neurone, which is also connected to a motor neurone. These then send a signal to an effector (such as your biceps muscle) that causes your muscle to contract to move away from the hot tray to stop you from burning yourself.
Sometimes, your conscious brain can send signals down to the relay neurones in the spinal cord to inhibit (stop) this reflex and allow you to withstand the heat. Although this probably isn't a good idea... But it makes great shows like in firewalking! Although this also involves other nifty tricks to do with water.